Dynamics: Impact of Rigid Bodies

Analyzing collisions where objects rotate requires moving beyond particle dynamics to account for both translational and rotational effects. Whether designing a safer vehicle, optimizing a manufacturing process, or understanding sports equipment, mastering rigid body impact gives you the tools to predict complex post-collision motion and energy dissipation. This analysis is crucial for engineering systems where rotation significantly influences the outcome, from gear teeth clashing to a satellite docking in orbit.

Impulsive Force Analysis

The foundation of impact analysis is the concept of an impulsive force, an extremely large force acting over an infinitesimally short time interval. During impact, non-impulsive forces like weight, spring forces, or friction are negligible because their impulse (force multiplied by time) is vanishingly small compared to that of the contact force. This principle allows us to analyze the collision in isolation.

The linear impulse-momentum principle states that the linear impulse acting on a body equals the change in its linear momentum. For a rigid body, this is: $$\int \mathbf{F}dt=m{\mathbf{v}}_G^{\prime }-m{\mathbf{v}}_G$$ Here, $\mathbf{F}$ is the impulsive force, $m$ is the mass, and ${\mathbf{v}}_G$ and ${\mathbf{v}}_G^{\prime }$ are the velocities of the center of mass $G$ before and after impact, respectively. Similarly, the angular impulse-momentum principle relates the angular impulse about a point (often $G$ or a fixed point) to the change in angular momentum. About the center of mass, this is: $$\int \mathbf{r}\times \mathbf{F}dt=I_G{\mathbf{\omega }}^{\prime }-I_G\mathbf{\omega }$$ Here, $\mathbf{r}$ is the position vector from $G$ to the point of impulse application, $I_G$ is the mass moment of inertia about $G$, and $\mathbf{\omega }$ and ${\mathbf{\omega }}^{\prime }$ are the angular velocities before and after impact. These two vector equations provide three scalar equations (in planar motion) to solve for unknowns.

Angular Velocity Change and Impact at Arbitrary Points

The point where the impulsive force acts is critical. If the line of action of the impulsive force passes through the body's center of mass $G$, then $\mathbf{r}=0$ and the angular impulse about $G$ is zero. Consequently, the angular velocity does not change (${\mathbf{\omega }}^{\prime }=\mathbf{\omega }$), and only the translational velocity is affected—the collision behaves like a particle impact.

However, impact at arbitrary points on a rotating body is the general and more complex case. Here, $\mathbf{r}$ is non-zero, creating an angular impulse that changes the body's spin. You must account for both the change in linear velocity of $G$ and the change in angular velocity $\mathbf{\omega }$. Consider a rotating rod that strikes a fixed pin. The impulsive force from the pin creates an angular impulse about $G$, causing an abrupt, often counterintuitive, change in the rod's rotation rate. The key is to simultaneously apply the linear and angular impulse-momentum equations, recognizing that the impulsive force appears in both.

Coefficient of Restitution for Rigid Body Impact

The coefficient of restitution, $e$, quantifies the elasticity of the collision. For particles, it is defined as the ratio of the relative speed of separation to the relative speed of approach along the line of impact. For rigid bodies, this definition still applies, but you must be meticulous about which points you use to calculate the relative velocity.

You must apply the coefficient of restitution equation to the points of contact on each body, in the direction normal to the impact surface. For two bodies A and B, let their contact points be $C_A$ and $C_B$. The velocities of these points are found from the center-of-mass velocity and the angular velocity: ${\mathbf{v}}_C={\mathbf{v}}_G+\mathbf{\omega }\times {\mathbf{r}}_{G\to C}$. The restitution equation is: $$e=\frac{({\mathbf{v}}_{C_B}^{\prime }-{\mathbf{v}}_{C_A}^{\prime })\cdot \mathbf{n}}{({\mathbf{v}}_{C_A}-{\mathbf{v}}_{C_B})\cdot \mathbf{n}}$$ where $\mathbf{n}$ is the unit normal vector to the impact surface. This equation provides the critical relationship between the normal components of velocity before and after impact, introducing the energy loss characteristic of plastic ($e=0$) or partially elastic ($0<e<1$) collisions.

Energy Loss Calculations

With the exception of a perfectly elastic impact ($e=1$), kinetic energy is not conserved. Calculating energy loss is vital for understanding damping, wear, and deformation. The total kinetic energy $T$ of a rigid body in planar motion is the sum of its translational and rotational parts: $$T=\frac{1}{2}m{v}_G^2+\frac{1}{2}{I}_G{\omega }^2$$ To find the energy lost $\Delta T$ during an impact, you compute the total kinetic energy of the system (all bodies involved) immediately before impact $T_1$ and immediately after impact $T_2$: $$\Delta T=T_1-T_2$$ This loss appears as heat, sound, and permanent deformation. For instance, in a perfectly plastic impact ($e=0$), the maximum possible kinetic energy is dissipated, often leading to the bodies moving together post-collision. The energy loss is directly tied to the coefficient of restitution; a lower $e$ results in greater $\Delta T$.

Ballistic Pendulum Analysis

The ballistic pendulum is a classic application that combines rigid body impact with work-energy principles to measure projectile velocity. The system consists of a pendulum bob (a block of mass $M$ and moment of inertia $I$) that is struck by a projectile (mass $m$) with initial velocity $v$. The analysis occurs in two distinct phases.

Phase 1: Impact. The projectile embeds itself in the bob (a perfectly plastic impact, $e=0$). During this infinitesimally short time, the pendulum support provides an impulsive reaction force, but if we take angular impulse-momentum about the fixed pivot point O, this external impulse has zero moment arm. Thus, angular momentum about O is conserved. Writing this conservation equation allows you to solve for the angular velocity ${\omega }^{\prime }$ of the pendulum (with embedded projectile) immediately after impact: $$mvd=(I_O+m{d}^2+M{r}_G^2){\omega }^{\prime }$$ Here, $d$ is the impact distance from O, and $I_O$ is the pendulum's moment of inertia about O.

Phase 2: Swing. After impact, the pendulum swings upward. During this phase, non-impulsive forces (gravity) do work. Applying the conservation of energy principle from just after impact (kinetic energy) to the maximum height (potential energy) gives: $$\frac{1}{2}{I}_{O,\text{total}}({\omega }^{\prime }{)}^2=(m+M)gh$$ By measuring the height $h$, you can back-solve for ${\omega }^{\prime }$, and then substitute into the angular momentum equation from Phase 1 to find the original projectile velocity $v$. This two-stage model perfectly separates the impulsive event from the subsequent conservative motion.

Common Pitfalls

1. Applying restitution to the center of mass: A frequent error is using the velocities of the centers of mass in the coefficient of restitution equation. This is incorrect. You must always apply it to the velocities of the points of contact on each body along the line of impact. Using $v_G$ instead of $v_C$ will yield a wrong result unless the impact is central (through $G$).
2. Ignoring the impulse at a fixed pin or support: During the impact phase of a ballistic pendulum or any body pivoted at a point, there is an external impulsive force at the pivot. While its linear impulse affects the linear momentum equation, you can often avoid it by wisely choosing to conserve angular momentum about the pivot point, where the impulse has zero moment and thus creates no angular impulse.
3. Assuming kinetic energy is conserved: Unless explicitly stated that an impact is perfectly elastic ($e=1$), you should never assume kinetic energy is conserved. Most real-world engineering impacts are inelastic, and energy loss calculations are a primary outcome of the analysis. Confusing momentum conservation (which always holds for the system if no external linear impulse acts) with energy conservation is a fundamental mistake.
4. Neglecting sign conventions in restitution: The coefficient of restitution equation $e=(v_B^{\prime }-v_A^{\prime })/(v_A-v_B)$ is scalar but depends on a defined positive direction (typically the line of impact, $\mathbf{n}$). You must assign a positive direction and consistently apply it to all velocity components. Mixing signs will lead to an incorrect value for $e$ or the post-impact velocities.

Summary

• Impulsive force analysis isolates the collision using linear and angular impulse-momentum principles, allowing you to solve for sudden changes in velocity.
• Impact at an arbitrary point on a rotating body changes both its translational and angular velocity, governed by the simultaneous application of linear and angular impulse-momentum equations.
• The coefficient of restitution ($e$) is applied to the normal components of the relative velocity at the points of contact, not the centers of mass, determining the collision's elasticity.
• Energy loss $\Delta T$ is calculated from the difference in total kinetic energy before and after impact and is significant in all inelastic collisions.
• Ballistic pendulum analysis elegantly separates the problem: use angular momentum conservation about the pivot during the plastic impact phase, then use energy conservation for the subsequent swing to determine initial projectile velocity.